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Can you give me just a hint to calculate the following limit? $$\lim_{n\to\infty}\frac{1}{(\ln n)^{\ln n}}$$ I am trying using the squeeze rule without success. I tried also l'Hopital but the limit become tougher.

Thank you.

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    $\begingroup$ Take the logarithm. $\endgroup$ – Martín-Blas Pérez Pinilla Jan 26 '14 at 19:04
  • $\begingroup$ @Charlie Please, instead of writing $->$ by typing ->, write $\to$ by typing \to or \rightarrow. $\endgroup$ – user93957 Jan 26 '14 at 19:04
  • $\begingroup$ @Aðøbe Ok, I'm sorry. I'll do it from now on. $\endgroup$ – Charlie Jan 26 '14 at 19:08
  • $\begingroup$ @Charlie No problem! $\endgroup$ – user93957 Jan 26 '14 at 19:11
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$\ln(n) \rightarrow \infty$ as $n \rightarrow \infty$. So, $\ln(n)^{\ln(n)} \rightarrow \infty$ as $n \rightarrow \infty$. Hence the limit is $0$.

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  • $\begingroup$ Please, write $\ln$ by typing \ln instead of $ln$ made by typing ln. $\endgroup$ – user93957 Jan 26 '14 at 19:06
  • $\begingroup$ ok- will do. Thanks @Aðøbe $\endgroup$ – voldemort Jan 26 '14 at 19:07
  • $\begingroup$ You're welcome! ;-) $\endgroup$ – user93957 Jan 26 '14 at 19:08
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    $\begingroup$ @Charlie: No- it is infinite. $\endgroup$ – voldemort Jan 26 '14 at 19:11
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    $\begingroup$ @Charlie $\infty^\infty$ is not an indeterminate form. $\endgroup$ – user93957 Jan 26 '14 at 19:14

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